L(s) = 1 | − 2-s + 4-s − 5-s + 4·7-s − 8-s + 10-s − 6·13-s − 4·14-s + 16-s − 2·17-s + 19-s − 20-s − 8·23-s + 25-s + 6·26-s + 4·28-s − 2·29-s − 8·31-s − 32-s + 2·34-s − 4·35-s + 10·37-s − 38-s + 40-s − 6·41-s − 8·43-s + 8·46-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.51·7-s − 0.353·8-s + 0.316·10-s − 1.66·13-s − 1.06·14-s + 1/4·16-s − 0.485·17-s + 0.229·19-s − 0.223·20-s − 1.66·23-s + 1/5·25-s + 1.17·26-s + 0.755·28-s − 0.371·29-s − 1.43·31-s − 0.176·32-s + 0.342·34-s − 0.676·35-s + 1.64·37-s − 0.162·38-s + 0.158·40-s − 0.937·41-s − 1.21·43-s + 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.843934774420797448209817879143, −8.015175707387893841810959094794, −7.63455573047029776010887453716, −6.85863386611528831207553810432, −5.60838150051347997707141447038, −4.82724474778665646545710325876, −3.95662118047845736908909485900, −2.48362521318337386400401120703, −1.67101905332336168240536023069, 0,
1.67101905332336168240536023069, 2.48362521318337386400401120703, 3.95662118047845736908909485900, 4.82724474778665646545710325876, 5.60838150051347997707141447038, 6.85863386611528831207553810432, 7.63455573047029776010887453716, 8.015175707387893841810959094794, 8.843934774420797448209817879143